It is known [1]}, [2]}, [3]} that the classical KR field strength terms \(H^2\)  in (REF ) can be absorbed into a generalised curvature with a “torsionful connection (Christoffel symbol)”, with the contorsion proportional to the field strength \(H_{\mu \nu }^\rho \)  ,
\({\overline{\Gamma }}_{\mu \nu }^{\rho } = \Gamma _{\mu \nu }^\rho + H_{\mu \nu }^\rho \ne {\overline{\Gamma }}_{\nu \mu }^{\rho }\) ,
where \(\Gamma _{\mu \nu }^\rho = \Gamma _{\nu \mu }^\rho \)  is the torsion-free Christoffel symbol. In the string effective action this is only an approximation since there are higher order corrections in \(\alpha ^{\prime } \) . Nonetheless it is a useful starting point.This is our motivation to consider fermions on a torsional manifolds in a microscopic model for our background [4]}, [5]}, [6]}, [7]}, [8]}. Lorentz invariance does not hold in the presence of a torsion background.
